Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/232

 {{MathForm2|(50)|$$\left.\begin{array}{rl} x= & \lambda+\mu+\nu-c-b,\\ \\y^{2}= & 4\frac{(b-\lambda)(\mu-b)(\nu-b)}{c-b},\\ \\z^{2}= & 4\frac{(c-\lambda)(c-\mu)(\nu-c)}{c-b};\end{array}\right\} $$}}

{{MathForm2|(51)|$$\left.\begin{array}{l} \lambda=\frac{1}{2}(b+c)-\frac{1}{2}(c-v)\cos h\alpha,\\ \mu=\frac{1}{2}(b+c)-\frac{1}{2}(c-b)\cos\beta,\\ \nu=\frac{1}{2}(b+c)+\frac{1}{2}(c-b)\cos h\gamma;\end{array}\right\} $$}}

{{MathForm2|(52|$$\left.\begin{array}{l} x=\frac{1}{2}(b+c)+\frac{1}{2}(c-b)(\cos h\gamma-\cos\beta-\cos h\alpha).\\ \\y=2(c-b)\sin h\frac{\alpha}{2}\sin\frac{\beta}{2}\cos h\frac{\gamma}{2},\\ \\z=2(c-b)\cos h\frac{\alpha}{2}\cos\frac{\beta}{2}\sin h\frac{\gamma}{2}.\end{array}\right\} $$}}

When $$b=c$$ we have the case of paraboloids of revolution about the axis of $$x$$, and

The surfaces for which $$\beta$$ is constant are planes through the axis, $$\beta$$ being the angle which such a plane makes with a fixed plane through the axis.

The surfaces for which $$\alpha$$ is constant are confocal paraboloids. When $$\alpha=0$$ the paraboloid is reduced to a straight line terminating at the origin.

We may also find the values of $$\alpha,\ \beta,\ \gamma$$ in terms of $$r,\theta$$ and $$\phi$$, the spherical polar coordinates referred to the focus as origin, and the axis of the parabolas as axis of the sphere,

We may compare the case in which the potential is equal to $$\alpha$$, with the zonal solid harmonic $$r_i Q_i$$. Both satisfy Laplace’s equation, and are homogeneous functions of x, y, z, but in the case derived from the paraboloid there is a discontinuity at the axis, and $$i$$ has a value not differing by any finite quantity from zero.

The surface-density on an electrified paraboloid in an infinite field (including the case of a straight line infinite in one direction) is inversely as the distance from the focus, or, in the case of the line, from the extremity of the line.